In this paper, we propose and study a new notion of local error bounds for a convex inequalities system defined in terms of a minimal time function. This notion is called generalized local error bounds with respect to F, where F is a closed convex subset of the Euclidean space \({\mathbb {R}}^n\) satisfying \(0\in F\). It is worth emphasizing that if F is a spherical sector with the apex at the origin then this notion becomes a new type of directional error bounds which is closely related to several directional regularity concepts in Durea et al. (SIAM J Optim 27:1204–1229, 2017), Gfrerer (Set Valued Var Anal 21:151–176, 2013), Ngai and Théra (Math Oper Res 40:969–991, 2015) and Ngai et al. (J Convex Anal 24:417–457, 2017). Furthermore, if F is the closed unit ball in \({\mathbb {R}}^n\) then the notion of generalized local error bounds with respect to F reduces to the concept of usual local error bounds. In more detail, firstly we establish several necessary conditions for the existence of these generalized local error bounds. Secondly, we show that these necessary conditions become sufficient conditions under various stronger conditions of F. Finally, we state and prove a generalized-invariant-point theorem and then use the obtained result to derive another sufficient condition for the existence of generalized local error bounds with respect to F.

在本文中，我们提出并研究了根据最小时间函数定义的凸不等式系统的局部误差界的新概念。这个概念称为关于F的广义局部误差边界，其中F是满足\（0 \ in F \）的欧几里得空间\（{\ mathbb {R}} ^ n \）的闭合凸子集。值得强调的是，如果F如果是一个球形的扇形体，其顶点在原点，那么这个概念就成为一种新型的方向误差范围，它与Durea等人的几个方向规则性概念密切相关。（SIAM J Optim 27：1204–1229，2017），Gfrerer（设定值Var Anal 21：151–176，2013），Ngai和Théra（Math Oper Res 40：969–991，2015）和Ngai等人。（J Convex Anal 24：417-457，2017）。此外，如果F是\（{\ mathbb {R}} ^ n \）中的闭合单位球，则关于F的广义局部误差范围的概念简化为通常的局部误差范围的概念。更详细地说，首先，我们为这些广义局部误差范围的存在建立了几个必要条件。其次，我们证明了这些必要条件在F的各种更强条件下变为充分条件。最后，我们陈述并证明广义不变点定理，然后使用获得的结果来推导关于存在关于F的广义局部误差界的另一个充分条件。